430 SWAN. 



The velocity potential at a distance r from a given point in a medium 

 vibrating with simple harmonic motion may be expressed as 



P cos (2-irnt + e) 



where n is the pitch of the sound, and P and e are independent of the 

 time /, but are dependent on ;•. This may be represented by the real 

 part of the imaginary expression 



v _ p e i(kvt+t) 



where 



. _ 2irn 2tt 



v being the velocity of sound and X the wave length. 

 The equation of motion has the general form 



d 2 <p 



but 



w - » ! vv = * 



T7 =• - A-VPe , '( fcw+t - > 



7.'' ,.'' 



hence 



vv + />-v + r 2 * = 



19 



The solution of this equation is 



-,-ikr 



1 C C Ce- ikr 

 wr J J J r 



Air 



When the force has the value <t>i and is distributed only over a surface 

 S in the medium. 



1 /' fe- ikr 



dS 



d<p . 

 If tlic surface is plane, and if ^ — is the normal velocity of the medium 



an 



at the surface of the plane, then 



* ' " 2^ J J dn r 



-ikr 



dS 



19 Rayleigh, Theory of Sound, Art. 277, (3). 



